Ebook: Mathematical Statistics with Applications

In Spring 2010 I used this book to teach a calculus-based undergraduate course in statistics and probability at Metropolitan State University in Saint Paul. I covered chapters 1, 2, 3, 4, 6, 7, 12 and 13.

In short, I cannot recommend the book. Explanations are often verbose and hard to follow, there are no problem solutions in the back of the book, and there is a plethora of typos and more serious mistakes throughout the book. Below is a detailed list of the mistakes I found in the first printing of the first edition. It is to be hoped that these will be corrected in later printings.

The numbers of many problems haven't been checked, often leading to outrageous and unrealistic probabilities or p-values etc. Examples are the problems 4.1.8, 4.1.9, 4.1.10, 4.1.13, 4.1.14, 4.4.5, 4.4.6, 7.1.6, 7.4.8, 7.4.12, 7.4.15, 7.5.5, 7.6.3, 7.6.4, 12.4.7.

P.8 Definition 1.3.1 falsely asserts that if every element of the population has the same chance of being chosen, then every size n subset is equally likely.

P.27 Definition 1.5.2 for upper and lower quartile is incorrect (or at least unclear) in the case of several identical data items.

P.67 Birthday paradox example, final answer for n=40 is wrong.

P.73 Example 2.4.3 notation and concept of "genetic makeup" and "allele" are not sufficiently explained for non-biologists.

P.77 Step three of Bayes' Rule: "denominator probability" is not explained; the rule as written is not clear.

P.83 Problem 2.6.29 Variable i is used twice. "Head ith probability p" is unclear.

P.85 Line 5: "for all real values x of X" should be "for all real numbers"; this is the whole point of the sentence.

P.90 Problems 2.5.2 and 2.5.3 use the terms "probability function" and "distribution"; should be "probability mass function". 2.5.4 should not use the notation p(x) for the cdf.

P.91 Problem 2.5.5 No random variable can have the given cdf, since it isn't right continuous.

P.95 Definition 2.6.3 The crucial difference between E((X-mu)^2) and (E(X-mu))^2 is obscured by writing E(X-mu)^2. The definition is not clear. Ambiguous notations such as E(X-mu)^2, EX^2, E(X-my)^3 and E(X-muX)(Y-muY) are used throughout the book.

P.106 Problem 2.6.8: variance 4 is not needed.

P.121 The Poisson approximation "rule of thumb" np<5 is only valid for large n.

P.128 Explanation of QQ plot is not intelligible. r_i is not "the rank i-th observation", but the "rank of the i-th observation". Figure 3.3 cannot be understood; there should be just as many points to the right of 0.0 as there are to the left. Furthermore, on p.312 the horizontal and vertical axes of QQ plots are reversed. On p.222 it is called "normal-score plot", using a different rule. Step 4 (45 degree line) is wrong.

The fundamental relationship of Poisson variables and exponential variables in describing Poisson processes is nowhere mentioned.

P.141 last line: the very first double integral in the book is wrong, confusing students immensely. "dxdy" should be "dydx". On p.142 bottom, "dxdy" is missing altogether.

P.146 Definition 3.3.5 E(X,Y) should be E(XY).

P.147 Theorem 3.3.2(b) Var(X|Y) has not been defined.

P.149 top: rho does not depend on lower case x and y, so rho(x,y) does not make sense. The statement "if Y=aX+b (a<>0) then rho=1" is false. In the "Properties" box, Cov(X;Y) must be replaced by rho.

The crucial theorem stating that the variance of a sum of independent variables equals the sum of the variances is used throughout the book, but never clearly stated or proved.

P.151 Problem 3.3.5 These are marginal probability mass functions, not density functions.

P.152 Problem 3.3.14 The notation for "proportional" has not been introduced.

P.153 Problem 3.3.18 X and Y cannot be independent.

P.165 line 6: restriction K>=2 not needed. However it should be mentioned that Chebyshev's Theorem is only interesting for K>1.

P.166 Theorem 3.5.2 The notations X-bar and S_n have not been defined. In the proof, the undefined abbreviation "iid" is used (which also cannot be found in the index). However, the variables were not assumed to be iid.

P.168 Theorem 3.5.2 It should be emphasized that the central limit theorem has one important assumption: that the mean and variance be finite. On the other hand, the moment generating function is not needed for the statement.

P.169 Example 3.5.7 (b) assumption n=300 is missing. (c) what is n?

P.170 Example 3.5.8 is completely unclear. "such that they can be confident"? How confident? In the solution the confidence 0.99 appears out of the blue. The maximum of pq should be explained with calculus. The fundamental importance of this example needs to be explained: it is used to estimate any unknown probability, not just unfair coins.

P.172 Problem 3.5.4 This should be "number of cars arriving at a busy intersection in a given 20 minute interval".

P.173 All the problems from section 3.5 return later in section 4.1.

P.184 line 5: the normal distribution is not called "the theorem of de Moivre-Laplace".

P.185 Example 4.1.1 The last paragraph should not be part of the example.

P.186 Theorem 4.1.1 Was already used on p.167. The last sentence is Theorem 3.5.2.

P.187 Example 4.1.2 Cannot be solved at this point, needs Corollary 4.2.2!

P.213 "z-series" should be "z-scores"

P.215 "(a+0.5)" should be "(b+0.5)"

P.217 Problem 4.4.10 "90 heads" should be "at least 90 heads".

P.316 line 2: "we know that" should give a reference to the proper theorem.

P.328 Problem 6.5.9 "1997" and "1998" should be "2000" and "2001".

P.341 figure at the bottom: explain the meaning of the two distributions, "critical value" has not been explained.

Many hypothesis tests are taken to be one-tailed, without proper justification. In the real world, such justification is extremely rare.

While the text emphasizes the importance of computing p-values, the worked examples almost never show the computation of p-values.

The crucial difference between the size of an effect and the statistical significance of an effect is not sufficiently emphasized.

P.345 Example 7.1.5 is unrealistic, testing a null hypothesis of mu=15 against an alternative of mu=16. The value mu=16 should only be used to compute beta, not as alternative hypothesis.

P.346 The example is unintelligible, since mu_a is not explained.

P.349 Problem 7.1.5(c) beta=0.8 is completely unrealistic.

P.365 Top: Assumption n>=30 for the large sample case is incomplete: we also need finite variance!

P.373 Line 4: X_{1n_1}, X_{2n_2}

P.387 Problem 7.5.9: Cannot assume that the samples are independent; only part (c) makes any sense.

P.389 Top: Definitions of Q^2 is wrong; denominator should be np_i.

P.390 Box: "exact methods are available etc." is not an assumption.

P.390 Should point out the crucial logical issue here: if Q^2 is small, then we don't have enough evidence to reject H_0; this DOES NOT imply that we have evidence in favor of H_0! Same mistake in example 7.6.3.

P.391 "n=4" should be "k=4"; n is 500.

P.392 Section 7.6.2 Horrible language.

P.393 bottom of box: E_{ij} = n_{i.}n_{.j} (Same typo on p.394).

P.393 Explanation for the degrees of freedom formula (r-1)(c-1) is missing.

P.398 Problem 7.6.5 Where does H_0 come from? Does it talk about before or after advertising?

P.601 Line 6: "differing only in median or mean 3"??

P.602 The given approach for computing confidence intervals for the median is wrong. The median will be below X_{a+1} with probability alpha/2.

P.603 Example 12.2.1. Don't report this as a 95% confidence interval; state the correct confidence level instead.

P.607 Section 12.3.1 Assumption of continuous distribution should be stated first; no need to repeat P(X<=M)=0.5 later.

P.608 Bottom. Explain what "ties" are.

P.621 Step 2 of box is not clear.

P.621 Bottom: why switch from N_{1b} to N_{1a}?

P.622 Box: Variance formula contradicts the one given on the previous page; expected value formula should have n_1, not N_1.

P.625 Line 6: denominator should be sqrt(1.68), not sqrt(16,900).

P.685 Metropolis Algorithm: the problem that the algorithm solves is not motivated. Surely it should be trivial to generate values according to a given discrete probability distribution? After all, step 1 of the algorithm requires just that.

P.659 Why are "pseudo values" defined if they are not used for the jackknife estimate?

P.666 Example 13.3.2 Last sentence: 12.37 and 43.9 should not be compared: one is a standard error, the other a standard deviation. Need to divide by square root of 12.

P.671 line 5: g(x,y)|theta should be g(x,y|theta)

P.663 The word "resampling" is used, but the core idea of resampling is not explained.

P.684 line 1: state space S is not used.

P.684 line 8: p(x,y) = p_{m,m+1}(x,y)

P.684 "...then the limiting distribution is the invariant distribution" does not make sense: they are always the same!

P.684 "ergodic" is not defined.

P.689 Metropolis-Hastings algorithm: "x_n = x_{n-1}" should be "x_{k+1}=x_k".

P.758 Appendix III: In Def 3.2.7, the exponential distribution was given with a parameter